Coupled Inductors, Mutual Inductance & Dot Convention

Two coils are coupled when part of the flux produced by one links the other. Their terminal equations can no longer be written independently: each voltage contains a self-inductance term and a mutual-inductance term.

Coupled equations #

If both reference currents enter dotted terminals and both voltages are positive dot-to-undotted:

$$v_1=L_1\frac{di_1}{dt}+M\frac{di_2}{dt},$$

$$v_2=M\frac{di_1}{dt}+L_2\frac{di_2}{dt}.$$

Reverse exactly one current reference or one voltage polarity and the mutual terms change sign. In phasor form, replace each derivative by $j\omega$.

The mutual inductance is bounded by

$$M=k\sqrt{L_1L_2},\qquad 0\le k\le 1.$$

$k=1$ is ideal coupling with no leakage flux in the model; practical magnetic parts have $k<1$.

Reading the dots #

The rule is relational: a current entering the dotted terminal of one winding induces a positive voltage at the dotted terminal of the other. Dots do not identify absolute positive terminals and do not imply the windings are physically connected.

For series-connected coils:

$$L_{aiding}=L_1+L_2+2M,$$

$$L_{opposing}=L_1+L_2-2M.$$

Which equation applies follows from the dot connections and current direction, not from whether the coils are drawn side by side.

Model limits #

The $L_1,L_2,M$ model captures linear magnetic coupling. Real parts may also require winding resistance, leakage inductance, core loss, magnetizing nonlinearity, saturation, capacitance, and frequency-dependent loss. Analog Devices' coupled-inductor examples show why a model calibrated only to nominal inductance can disagree with the bench when coupling and leakage are omitted.

Reversing one current reference or one voltage polarity flips the corresponding mutual signs. It does not change the physical coupling.